gapMinusOne_at_D3

formal source

  93/-- The integer `D²(D+2) - 1`. At `D = 3` this equals `44`. -/
  94def gapMinusOne (d : ℕ) : ℕ := integrationGap d - 1
  95
  96theorem gapMinusOne_at_D3 : gapMinusOne D = 44 := by native_decide
  97
  98theorem gapMinusOne_factor : gapMinusOne D = 4 * 11 := by native_decide
  99
 100/-! ## φ-power identity (matter-balance bridge) -/
 101
 102noncomputable section
 103
 104/-- The active edge count per fundamental tick. -/
 105def A : ℤ := 1
 106
 107/-- The φ-power balance identity at `D = 3`:
 108    `φ^(A − gap) · φ^gap = φ`, equivalently `φ^(−44) · φ^45 = φ`. -/
 109theorem gap_balance :
 110    phi ^ (A - ↑(integrationGap D)) * phi ^ (↑(integrationGap D) : ℤ) = phi := by
 111  have hg : (↑(integrationGap D) : ℤ) = 45 := by exact_mod_cast integrationGap_at_D3
 112  rw [hg, show A = (1 : ℤ) from rfl, ← zpow_add₀ (ne_of_gt phi_pos)]
 113  have : (1 : ℤ) - 45 + 45 = 1 := by norm_num
 114  rw [this, zpow_one]
 115
 116end
 117
 118/-! ## Master certificate -/
 119
 120structure IntegrationGapCert where
 121  config_dim : configDim D = 5
 122  parity_count : parityCount D = 9
 123  gap_value : integrationGap D = 45
 124  gap_minus_one : gapMinusOne D = 44
 125  coprime_at_3 : Nat.Coprime (2 ^ D) (integrationGap D)
 126  odd_coprime : ∀ k, Nat.Coprime (2 ^ (2*k+1)) ((2*k+1)^2 * (2*k+3))